The
Geostrophic Assumption
The Horizontal Equation of Motion and the
NavierStokes Equation can be simplified for large scale flow in the atmosphere
and the ocean with the assumption that friction and viscosity effects are small
on an order of magnitude basis.
In that case, the horizontal frictionless equation of motion in concept
form reduces to:
Total Acceleration of (Air/Water) Parcel = Pressure Gradient Acceleration and Coriolis Acceleration
Geostrophic Wind Equation
The pressure gradient terms can be transformed by
use of the hydrostatic equation into height gradient terms. This allows the magnitude of the speed to be
related to height gradients. We’ll do
that transformation in ERTH 465. For
now, we simply are inserting those expressions into the Horizontal Equation of
Motion in natural coordinates.
(1)
Let’s apply this equation to the case we
examined in the last class, as shown in Figure 1 below.
Figure 1: Balance of forces or accelerations in the
horizontal Equation of Motion for the case of an initially stationary air
parcel at the 500 mb level in the middle latitudes. A_{1}
indicates the initial state and A_{2}
the final state, as discussed in class.
At the outset, the parcel shown at A_{1}
is at rest with respect to the surface of the earth and the pressure gradient
acceleration term acts northward. The
natural coordinate system is set up so that the positive s axis is tangent to
the acceleration vector with the positive n axis at a counterclockwise right
angle to it. On the right side of
Equation 1 there can be no Coriolis Acceleration since there is no motion, and
there is no height gradient along the n axis.
Hence, Equation (1) simplifies to
(2)
which is a mathematical equivalent of the
first rule of thumb with respect to motion we learned in this class: in the absence of other effects, air
accelerates from regions of higher values of pressure (in this case, heights)
to lower values, at right angles to the contours.
The final state we discussed in class is
shown at A_{2}. In this case we saw that the wind is blowing
parallel to height contours. Hence, there is no variation of height along the
wind streamlines, in that case, and ∆z/∆s is zero.
In addition, if air is moving parallel to the
isobars or height contours and there is a perfect balance between the pressure
gradient acceleration and the Coriolis acceleration, the total acceleration
experienced by the air parcel is zero, although the wind vector will be large. This will remain true as long as the field
of isobars or height contours is not changing (the pressure gradients do not
change).
In that restrictive circumstance, the net
acceleration on the left hand side of the expression is zero.
(3)
Solving (3) for the wind speed, V, termed the
"geostrophic wind" velocity, we get a relatively simple equation.
“Geostrophic” means in “earth balance” referring to the balance between
pressure gradient acceleration and Coriolis acceleration ultimately achieved by
wind and water parcels on the earth (but, in reality, in any atmosphere and on
any planet).
It provides insight into the flow that develops that is due to the balance of
the pressure gradient acceleration normal to the flow and Coriolis
acceleration. Because we made some
simplifications that are realistic only in certain circumstances, called the
geostrophic approximation, the wind symbol
obtained by solving for V in (3) normally has a subscript “g” to
indicate that these simplifications were made.
(4)
where f=. f is referred
to in a number of ways. In this case, it
is called the Coriolis parameter.
Equation (4) is known as the geostrophic
wind equation. It is elegant in its
simplicity and, because of that, students are very fond of it. It says that if the wind is geostrophic, at
a given latitude the wind speed is only related to the height (pressure)
gradient.
A Measure of How Realistic the Geostrophic
Assumption Is: The Rossby Number
It doesn’t take much consideration to realize
that the Geostrophic Assumption is unrealistic in many circumstances, many of
them obvious, and some subtle. In the
real atmosphere (and ocean) there are almost always small accelerations.
Notice in Figure 1 that the Coriolis
Acceleration vector increased as the wind speed increased. At the same time the total acceleration
decreased to, finally, zero in the last state.
In other words, the ratio of the total acceleration to Coriolis
acceleration gets smaller and smaller as the situation approaches geostrophic
balance.
This ratio is called the Rossby Number, and
can be used to decide to what extent the "geostrophic wind"
corresponds to the real wind, as a function of scale. Ratios of 0.1 or less
indicates that the total acceleration is one to two orders of magnitude smaller
than the Coriolis acceleration and, thus, the total acceleration can be dropped
on an order of magnitude basis: the wind flow will be tangent to the isobars or
height contours (see Table 1).
Values approaching 1 or more indicate that
the total acceleration experienced by an air parcel is very large, and cannot
be dropped out of the equation. In those cases, using the idea of the
geostrophic wind to explain what you see on a weather map will produce
confusion. In those cases, it will appear that air is moving at right angles to
the isobars from higher values of pressure to lower values of pressure. For example, the Rossby Number is very very
large the closer one gets to the Equator where Coriolis Acceleration is zero.
In Middle Latitudes (Middle Troposphere) Scale
of Circulation 
Ratio of Total
Acceleration to Coriolis Acceleration (Rossby Number) 
Real Wind Generally
Explained by Geostrophic Wind? 
10,000 km 
0.01 
Yes 
1,000 km 
0.1 
Yes 
100 km 
1.0 
Not really 
10 km 
3.0 
No 
1 km 
4.0 
No 
Table 1: Characteristic Rossby Number for different
scales of circulation and evaluated in the middle Troposphere. Red shading indicates that the geostrophic
assumption is invalid and air motion will be at right angles to isobars. Green shading indicates scales at which the
geostrophic wind assumption is valid and the geostrophic wind can be considered
a valid approximation of the real wind.
Another way to look at this is that real wind
is always made up of a portion that is geostrophic and a portion that is not
geostrophic (called AGEOSTROPHIC). Here's a simple algebraic expression that
summarizes this concept. In this case, the symbol, V, indicates the real (two
dimensional) horizontal wind in natural coordinates.
V = V_{geo} + V_{ageo } 
(3) 
It turns out that even at the synoptic and
macroscales, there are situations in which the total acceleration (ageostrophic
wind) is significant. These situations generally occur in certain portions, or
levels, of the troposphere. Also, there is a portion of the troposphere, or
level, in which the total acceleration (ageostrophic wind) is small. There the
wind will appear to be flowing parallel to the isobars (or height contours)
with a magnitude exactly determined by the pressure gradient. It also turns out that the ageostrophic
accelerations are related to the horizontal divergence patterns in the
atmosphere (see Table 2).
Level of the Troposphere

V_{geo} 
V_{ageo} 
Real Wind
"Looks" Like Geostrophic Wind? 
Net Acceleration 
Hori zontal Divergence 
Upper (300200 mb level) 
Yes 
Large 
Somewhat 
Large 
"Large" 
Middle (600
to 450 mb) 
Yes

None to
small 
Yes 
None to
small 
None to
small 
Lower (Sfc to 925 mb) 
Yes 
Large 
Somewhat 
Large 
"Large " 
Table 2:
Qualitative evaluation of the
presence of the geostrophic wind and the ageostrophic wind in the lower,
middle, and upper troposphere. Only in
the middle troposphere (green) is the geostrophic wind similar to the actual
wind observed. At the other levels ageostrophic
accelerations and winds are large contributions so that the actual wind cannot
be approximated by the geostrophic wind realistically.
Thus, students who embrace the concept of the
geostrophic approximation "love" the 500 mb chart, because there the
wind appears to flow parallel to the contours and in direct proportion to the pressure (or height)
gradient. That's the level at which the geostrophic approximation corresponds
most closely to the real wind.
This is not the case on 200 and 300
mb charts. It's true that even the
largest ageostrophic motions there are relatively small compared to the
geostrophic motion. So, at first glance it will appear that the wind is in
geostrophic balance. A closer look will reveal areas of cross contour flow,
and, also, areas in which the wind speeds do not match what one would expect
from solution of the geostrophic wind relation.
Finally, at the ground, there
is also substantial cross contour flow.
This "disruption" of the geostrophic wind balance occurs chiefly
because of friction. That's why the surface wind will most closely resemble the
geostrophic wind in regions in which frictional effects are minimal (e.g., over
oceans, over flat featureless plains).